Pawel M. Kozlowski

An effective method for generating nonadiabatic many‐body wave function using explicitly correlated Gaussian‐type functions

General formalism for the application of explicitly correlated Gaussian‐type basis functions for nonadiabatic calculations on many‐body systems is presented. In this approach the motions of all particles are correlated in the same time. The energy associated with the external degrees of freedom, i.e., the motion of the center of mass, is eliminated in an effective way from the total energy of the system. In order to achieve this, methodology for construction of the many‐body nonadiabatic wave function and algorithms for evaluation of the multicenter and multiparticle integrals involving explicitly correlated Gaussian cluster functions are derived. Next the computational implementation of the method is discussed. Finally, variational calculations for a model three‐body system are presented.

Implementation of analytical first derivatives for evaluation of the many-body nonadiabatic wave function with explicitly correlated Gaussian functions

A nonadiabatic many‐particle wave function is generated using an expansion in terms of explicitly correlated Gaussian‐type basis functions. In this approach, motions of all particles are correlated at the same time, and electrons and nuclei are distinguished via permutational symmetry. We utilize our newly proposed nonadiabatic variational approach [P. M. Kozlowski and L. Adamowicz, J. Chem. Phys. 95, 6681 (1991)], which does not require the separation of the internal and external motions. The analytical first derivative of the variational functional with respect to the nonlinear parameters appearing in the basis functions are derived and implemented to find the minimum. Numerical examples for the ground state of the hydrogen molecule are presented.

Newton–Raphson optimization of the many‐body nonadiabatic wave function expressed in terms of explicitly correlated Gaussian functions

A nonadiabatic many‐body wave function is represented in terms of explicitly correlated Gaussian‐type basis functions. Motions of all particles (nuclei and electrons) are treated equally and particles are distinguished via permutational symmetry. The nonadiabatic wave function is determined in a variational calculation with the use of the method proposed recently [P. M. Kozlowski and L. Adamowicz, J. Chem. Phys. 95, 6681 (1991)]. In this approach no direct separation of the center‐of‐mass motion from the internal motion is required. The theory of analytical first and second derivatives of the variational functional with respect to the Gaussian exponents and its computational implementation in conjunction with the Newton–Raphson optimization technique is described. Finally, some numerical examples are shown.

Application of explicitly correlated Gaussian functions for calculations of the ground state of the beryllium atom

The electronic energy of atoms and molecules may be evaluated accurately by the use of wave functions where the interelectronic distances are explicitly present. In particular, explicitly correlated Gaussian‐type functions make these types of calculations feasible and computationally tractable even for more extended systems. The resulting multielectron integrals may be reduced to standard one‐ and two‐electron integrals that are readily evaluated. Initial calculations have been made for the Be atom where all four electrons were correlated at the same time. The preliminary results show that accurate results may be obtained.

Newton-Raphson optimization of the explicitly correlated Gaussian functions for calculations of the ground state of the helium atom

Explicitly correlated Gaussian functions have been used in variational calculations on the ground state of the helium atom. The major problem of this application, as well as in other applications of the explicitly correlated Gaussian functions to compute electronic energies of atoms and molecules, is the optimization of the nonlinear parameters involved in the variational wave function. An effective Newton–Raphson optimization procedure is proposed based on analytic first and second derivatives of the variational functional with respect to the Gaussian exponents. The algorithm of the method and its computational implementation is described. The application of the method to the helium atom shows that the Newton–Raphson procedure leads to a good convergence of the optimization process.

Note on the density matrix for an arbitrary number of closed shells in a bare Coulomb field

An explicit relation between the density matrix and its s-state part is analyzed for electron closed shells moving in a bare Coulomb potential. The density matrix has a simple separable form in terms ofr1 +r2 and |r1 −r2|. It is demonstrated that for an arbitrary number of closed shells, the off-diagonal dependence is simply polynomial in the |r1 −r2| coordinate.